Mole Fraction · aiwiki.page

Mole fraction is a dimensionless quantity describing the composition of a mixture. It equals the amount of substance of a particular constituent divided by the total amount of all constituents. The International Union of Pure and Applied Chemistry uses amount fraction as a synonymous term. Because amounts of substance are proportional to numbers of specified entities, mole fraction also equals the corresponding number fraction. It describes relative composition rather than the quantity of material present. (goldbook.iupac.org)

Definition and notation

For constituent $i$ , the mole fraction is

x_i=\frac{n_i}{n_{\mathrm{tot}}} =\frac{n_i}{\sum_j n_j},

where $n_i$ is its amount, normally expressed in moles, and the denominator includes every constituent within the specified mixture. The symbol $x_i$ is conventional for liquid and solid mixtures; $y_i$ is often used for gaseous mixtures, particularly when liquid and vapor compositions must be distinguished. (goldbook.iupac.org)

For a nonempty mixture,

0\leq x_i\leq1,\qquad \sum_i x_i=1.

A pure substance has a mole fraction of one for its sole constituent. In a binary mixture, $x_2=1-x_1$ ; consequently, only one mole fraction is independently needed to specify its composition. For $k$ constituents, the normalization condition leaves $k-1$ independent composition variables. These relationships follow directly from the definition. (openstax.org)

Using the Avogadro constant, $n_i=N_i/N_{\mathrm A}$ , gives

x_i=\frac{N_i}{\sum_j N_j}.

The counted entities must be identified: they may be molecules, atoms, ions, or other specified elementary entities. A molecular fraction and an atomic fraction need not describe the same composition. (bipm.org)

Units and numerical expression

In the International System of Units, mole fraction has unit one because it is a ratio of quantities of the same kind. Its value can therefore be written simply as a number. Writing $\mathrm{mol/mol}$ explicitly can clarify that the ratio concerns amounts of substance rather than masses or volumes. Scaled forms such as $\mathrm{\mu mol/mol}$ and $\mathrm{nmol/mol}$ are also permitted. Thus, $400\ \mathrm{\mu mol/mol}$ represents a mole fraction of $4.00\times10^{-4}$ . (bipm.org)

Multiplying a mole fraction by 100 gives its numerical expression as a percentage. For example, $x_i=0.25$ corresponds to 25%. The description must still identify the underlying quantity: the same numerical percentage could otherwise refer to a mass or volume ratio. (bipm.org)

Calculation and other composition measures

As an illustrative calculation, a mixture containing $2.0$ mol of nitrogen and $1.0$ mol of oxygen has

x_{\mathrm{N_2}}=\frac23,\qquad x_{\mathrm{O_2}}=\frac13.

Doubling both amounts leaves these fractions unchanged. This illustrates that mole fraction specifies proportions, not sample size. (openstax.org)

When composition is given by mass, each constituent’s amount is calculated using its molar mass, $M_i$ . Substituting $n_i=m_i/M_i$ into the definition gives

x_i=\frac{m_i/M_i}{\sum_j m_j/M_j}.

Likewise, if $w_i$ denotes mass fraction, algebraic conversion yields

x_i=\frac{w_i/M_i}{\sum_j w_j/M_j}, \qquad w_i=\frac{x_iM_i}{\sum_j x_jM_j}.

Equal masses therefore do not generally imply equal mole fractions. (bipm.org)

Unlike molar concentration, which divides amount by solution volume, mole fraction contains no volume term. Unlike molality, which divides solute amount by the mass of the solvent, it includes the amounts of both solvent and solutes in its denominator. For a binary solution, substitution of the definitions gives

x_{\mathrm{solute}}=\frac{bM_{\mathrm{solvent}}} {1+bM_{\mathrm{solvent}}},

where $b$ is molality in $\mathrm{mol/kg}$ and $M_{\mathrm{solvent}}$ is in $\mathrm{kg/mol}$ . (openstax.org)

For fixed constituent amounts, mole fraction does not change merely because temperature or pressure changes. This differs from volume-based concentrations, which can change through expansion or compression. The qualification “fixed amounts” matters whenever material enters, leaves, or reacts. (openstax.org)

Gas mixtures and vapor–liquid equilibrium

For an ideal-gas mixture, Dalton’s law relates composition to partial pressure:

p_i=y_iP,

where $P$ is total pressure. Thus, a constituent with $y_i=0.20$ contributes one fifth of the total pressure. This pressure relationship assumes ideal-gas behavior; the definition of mole fraction itself does not require ideality. (openstax.org)

For an ideal solution, Raoult’s law connects liquid composition with equilibrium vapor pressure:

p_i=x_i p_i^{*},

where $p_i^{*}$ is the vapor pressure of pure constituent $i$ at the same temperature. With an ideal vapor, combining the two laws gives $y_iP=x_i p_i^{*}$ . Liquid and vapor mole fractions are therefore generally different, a distinction important in distillation. For a nonvolatile solute, Raoult’s law describes the reduction of solvent vapor pressure through the solvent’s mole fraction. (openstax.org)

Thermodynamic interpretation

In thermodynamics, composition affects chemical potential. For a liquid or solid mixture using a pure-component standard state,

\mu_i=\mu_i^{*}+RT\ln a_i, \qquad a_i=f_i x_i,

where $R$ is the gas constant, $a_i$ is thermodynamic activity, and $f_i$ is an activity coefficient. In an ideal mixture, $f_i=1$ , so activity equals mole fraction. In a nonideal mixture, mole fraction still records composition, while the activity coefficient accounts for departures from ideal thermodynamic behavior. These equations require a specified standard-state convention; mole fraction alone does not determine activity in every mixture. (goldbook.iupac.org)